Evaluation of Value-at-Risk Models During Volatility Clustering

Transcription

1 Evaluation of Value-at-Risk Models During Volatility Clustering An Empirical Study on the Financial Crisis of 2008 Master Programme in Finance Lund University Spring 2014 Author: Medjit Yalmaz Supervisor: Ph.D. Hans Byström

2 Abstract In the light of the financial crisis of 2008, risk management has become one of the most important topics in the financial world. This study applies five different VaR approaches, normal distribution, student s t distribution, historical simulation, age weighted historical simulation and volatility weighted historical simulation under three different sample windows. These parametric, non- parametric and semi- parametric approaches were applied on the historical closing prices of five highly diversified stock indices, OMXS 30, S&P 500, NIKKEI 225, DAX 30 and FTSE 100, where the focus has been on the period of Performance was evaluated by comparing the expected number of VaR breaks to the actual number of VaR breaks, the so called VaR ratio. The study found that most of the models using a larger sample window failed to cope with sudden changes in volatility, while the age weighted historical simulation seemed to cope well with sudden changes in market conditions in all sample windows. The study also found that forecasting volatility using EWMA in extreme market conditions failed to give accurate VaR estimates. 2

3 Acknowledgements I would like to take this opportunity to thank my supervisor Mr Ph.D. Hans Byström for his valuable guidance and comments throughout this process. I would also like to thank programme coordinator Mr Ph.D. Hossein Asgharian and all other teachers I have had the pleasure to have during my time at Lund University. 3

4 Table of Contents 1. INTRODUCTION BACKGROUND DEFINING VALUE- AT- RISK PURPOSE OF THE THESIS DELIMITATION 9 2. LITERATURE REVIEW ADVANTAGES OF VALUE- AT- RISK DRAWBACKS OF VALUE- AT- RISK PARAMETRIC APPROACH NORMAL DISTRIBUTION STUDENT S T- DISTRIBUTION NON- PARAMETRIC APPROACH HISTORICAL SIMULATION AGE WEIGHTED HISTORICAL SIMULATION SEMI- PARAMETRIC APPROACH VOLATILITY WEIGHTED HISTORICAL SIMULATION BACKTESTING VAR KUPIEC TEST CHRISTOFFERSEN TEST METHODOLOGY QUANTITATIVE APPROACH DEDUCTIVE APPROACH RELIABILITY AND VALIDITY THE INDICES OMX STOCKHOLM STANDARD & POOR S NIKKEI DEUTSCHER AKTIE INDEX FTSE DATA DESCRIPTIVE STATISTICS VAR ESTIMATES 26 4

5 3.7.1 NORMAL DISTRIBUTION STUDENT S T- DISTRIBUTION HISTORICAL SIMULATION AGE WEIGHTED HISTORICAL SIMULATION VOLATILITY WEIGHTED HISTORICAL SIMULATION EMPIRICAL EVIDENCE VAR UNDER NORMAL DISTRIBUTION BACKTEST OF VAR UNDER NORMAL DISTRIBUTION VAR UNDER STUDENT S T- DISTRIBUTION BACKTEST OF VAR UNDER STUDENT S T- DISTRIBUTION VAR UNDER HISTORICAL SIMULATION BACKTEST OF VAR UNDER HISTORICAL SIMULATION VAR UNDER AGE WEIGHTED HISTORICAL SIMULATION BACKTEST OF VAR UNDER AGE WEIGHTED HISTORICAL SIMULATION VAR UNDER VOLATILITY WEIGHTED HISTORICAL SIMULATION BACKTEST OF VAR UNDER VOLATILITY WEIGHTED HISTORICAL SIMULATION CHRISTOFFERSEN TEST OF INDEPENDENCE ANALYSIS CONCLUSION 50 BIBLIOGRAPHY 51 APPENDIX 55 5

6 1. Introduction 1.1 Background The 2008 financial crisis severely impacted financial markets and most importantly economies worldwide. Crashing stock prices with large corporations filing for bankruptcy on a daily basis, and government bailout programmes left the economies crawling on its knees. While the key drivers of the crisis were deep rooted and complex, there is a consensus among scholars and governments that failures in financial risk management was in fact one of the reasons behind the magnitude of the crisis (Sollis, 2009). Goldman Sachs CEO, Lloyd Blankfein, stated, while testifying in front of the Financial Crisis Inquiry Commission (FCIC), that: risk models, particularly those predicated on historical data were too often allowed to substitute for judgement (CSPAN, 2010). In the aftermath of the 2008 financial crisis, many studies have shown that Value- at- Risk (VaR) models couldn t predict nor forecast the magnitude of the financial collapse. Current market conditions, with inflated stock prices, high volatility and risk appetite, are showing similarities with the pre 2008 crisis conditions. The question is whether or not the financial world has learned from its mistakes or if it will hit again? Risk modelling and measurement is a unified leg in the operations of a financial institution and an important part of financial regulations. The last decades include several major and minor crises and regulatory changes are constantly appearing to address a better understanding of risk. While many other risk measurements exist, VaR overshadows the rest in usage and compatibility. VaR s practical superiority over its theoretical shortcomings is the main reason for the widely spread usage. Danielsson et al (2013) argues that this assumption is supported in most cases, both theoretically and empirically (Daníelsson et al., 2013). In its simplest form, VaR is used as an instrument to evaluate the losses arising from a potential decrease in the price of an asset. Simply, VaR quantifies the maximum occurred loss over a given time period and a given probability. In 1996 the Basel Committee incorporated VaR in the Basel I accord, continuing 6

7 with Basel II and III (Daníelsson et al., 2013). Financial firms are by law forced to engage in risk management stated by the Basel accord. The aim is to advocate financial stability by forcing financial institutions to hold enough capital reserves enabling them to minimize the risk of insolvency and default. Basel rules states that financial institutions must hold 8% of their risk- weighted assets as a capital reserve, also known as the capital ratio (Nilsson, 2013). The recognition of VaR by financial and regulatory commissions in recent times is confirming that the use of VaR as a risk measure is widely spread. The recent Basel III regulation advocates the use of these models (BIS, 2011), and the launch of J.P. Morgan s introduction of the RiskMetrics database for use with third- party VaR softwares explains the growing usage of VaR models by both financial and non- financial firms (Hendricks, 1996). A vast majority of parametric VaR models relies on the assumption that returns are normally distributed, and theoretically, this isn t something new in financial theory. Both the Black- Scholes option pricing formula and modern portfolio theory are based on the same assumption. Extensive research has shown that returns are seldom normally distributed but rather show signs of kurtosis and/or skewness (Mandelbrot, 1963). This leads to the fact that VaR models consistently miscalculate and in most cases underestimate the probability of high impact events. These VaR models depend excessively on the normal return distribution of the data sample at hand. The underestimation of high impact events was one of the reasons financial institutions were unable to react in a timely manner when the crisis of 2008 struck. Financial institutions generally apply a 99% confidence level when calculating VaR and should anticipate around VaR breaks (VB) a year if returns are normally distributed. Financial institutions, such as UBS, Credit Suisse and Morgan Stanley, have disclosed that they experienced 50, 24 and 18 VB s respectively during 2008 (Campbell, 2009). The objective of this study is to investigate different VaR approaches results during the recent financial crisis i.e. during volatility clustering in five stock indices. The choice of indices is based on three criteria, geographical location, ,99 = 2,52 7

8 high market capitalization and stock diversification. Daily closing prices of OMX Stockholm (OMXS 30), Standard & Poors 500 (S&P 500), NIKKEI 225, Deutscher Aktie IndeX (DAX 30) and FTSE 100 are used as the underlying data in this study. 1.2 Defining Value- at- Risk VaR is the smallest loss l such that the probability of a future portfolio loss! for an investor is larger than the loss l is less than or equal to 1!. VaR can be defined with the following mathematical equation:!"#!! =!"# l Pr! > l 1! 1.1 where! is a confidence level, e.g. 95% or 99%). VaR is, from a probabilistic view, the (1!) quantile of the return/loss distribution. Typical time periods used when determining VaR is h = 1 day or h = 10 days. The VaR equation may be rewritten in the following way under the presumption of a continuous loss distribution: Pr! >!"#!! = 1!. 1.2 This definition of VaR says that the probability of a loss! being larger than VaR is equal to 1!. 1.3 Purpose of the thesis The main objective of the thesis is to investigate the results of different parametric, non- parametric and semi- parametric VaR approaches during times of volatility clustering 2 (Dowd, 2002). The time period investigated will date from beginning of 1994 until end of 2013 and will therefore include the dot- com bubble of and the latest financial crisis of 2008, while the thesis will focus on the latter. There are no intentions to create a new approach to VaR, but rather to evaluate the behaviour of VaR models during a financial crisis. This will give a deep understanding of how financial institutions assess and how fast they can point out times of volatility clustering by using VaR as a risk measurement. Regulators are pushing for an increased level of control and extensive requirements on risk measurements with Basel III, commencing in 2018 as a 2 Volatility clustering occurs when financial returns show alternating moments/periods of high and low volatility i.e. if volatility is high (low) one day, it is expected to be high (low) the next day as well. 8

9 direct response to the recent financial crisis. It will be out of utter importance to evaluate the accuracy and shortcomings of VaR models, in order to judge whether or not VaR is capable of minimizing the risk of capital losses, and if the governmental support is justified. With this stated, the purpose of the thesis is therefore to evaluate the accuracy of a group of parametric, non- parametric and semi- parametric VaR models over the time period Different assumptions will be applied, including distributions and forecasting models, during the financial crisis of 2008 using historical closing prices of 5 different indices. 1.4 Delimitation Due to the limited time of writing the thesis certain delimitations are necessary. Including more than the chosen five stock indices would probably not increase the accuracy of the thesis materially since the chosen indices are well diversified, both in an industrial and geographical point of view. Though, other asset classes like foreign exchange (FX), rates, commodities, and fixed income (FI) has been excluded. This was done since most of the mentioned asset classes are traded over- the- counter (OTC) and each asset class has a numerous amount of different products. E.g. FX products include spot, forward/futures and swaps, where each product have a large amount of currency pairs to choose from. Therefore, due to the magnitude of different combinations of product types and the limited time of writing the thesis, these have been excluded. There are also some delimitations to which approaches have been selected to be analysed and tested. The thesis only focuses on VaR since it is still the most widely used model to estimate risk of a portfolio or an asset. Expected Shortfall (ES) is a slightly more sophisticated way of estimating risk, but given the lack of recognition this method has had at financial institutions, the objective is to evaluate VaR models. It is also important to stress that throughout the thesis, only 95% VaR has been taken in to consideration. Since the thesis focuses on relative short sample windows, it would be difficult to draw reliable conclusions tied to the purpose of the thesis using a higher confidence level. 9

10 2. Literature Review In this section of the thesis the advantages and drawbacks of VaR will be presented as a background. Furthermore the different parametric, non- parametric and semi- parametric approaches will be presented before the backtesting methods are discussed. 2.1 Advantages of Value- at- Risk There are two main advantages of using VaR as a risk measure. The first is that it gives a consistent risk measure across different asset classes and portfolios. In essence, it can measure and compare the risk of a fixed income portfolio with an equity portfolio. VaR gives a common view of the risk since it is measured in monetary value. Secondly, VaR estimates all types of risks and takes the correlation between the risk factors in to account. E.g. if there are two positions that by themselves are risky, but when combined in a portfolio, VaR could estimate a lower risk if they correlate negatively, and vice versa if the risk is highly correlated (Dowd, 2002). Dowd (2002) points out several practical ways of using VaR. (1) VaR can be used to set overall risk objectives and maintaining the risk appetite. (2) Since VaR gives an absolute figure on how risky a portfolio is, this figure can be used to determine capital allocation. (3) In the last 20 years, measuring and reporting VaR has become an important part for financial institutions in order to maintain and disclose their market risk e.g. in annual and quarterly reports. (4) Investment decisions can be made on the base of how VaR will change when pursuing an investment opportunity or when implementing hedging strategies. (5) It is also a way of managing the risk taking on trading books and is used to supervise traders. This is especially important in the light of Basel III in order for financial institutions to be within their limits and reducing risks. 2.2 Drawbacks of Value- at- Risk There are several drawbacks of the VaR model, one of them points out that estimates may be too far from the reality and causing imprecise risk, in essence making the risk estimates useless. Another worrying fact is that different VaR 10

11 approaches give significantly different results, which will be tested later in the thesis (Beder, 1995). These drawbacks clearly show the risk in using VaR as a risk measure, if VaR is inaccurate, and decisions are based fully on the base of VaR, investors may take on far more risk than what was originally expected (Hoppe, 1998). Another problem that is being stressed by Ju and Pearson (1999) is that if VaR is used to manage and supervise risk taking by traders. Traders will eventually be incentivised to seek positions where risk is over- or understated. Ju and Pearson (1999) show in their empirical results that the magnitude of VaR underestimations that rises from this behaviour is substantial. Taleb (1997) stresses that the widely spread use of VaR could impact financial markets. Since financial institutions constantly revises their positions and hedges due to changes in market prices, all players in a market might have the same behaviour, since they in the end rely on the same information. The result, or risk as you might call it, being that uncorrelated risk in the end becomes correlated, resulting in higher risk than what the VaR models might have suggested in the beginning. In the end, (1) VaR is silent about the loss of a VaR breach, i.e. how large the actual loss might be. The fact that (2) VaR is widely used, and in point of a financial crisis, everyone in the market will run for the fire exits at the same time. (3) It is not coherent, i.e. it does not always encourage diversification of the portfolio. The largest drawback may however be that (4) VaR is sensitive to incorrect assumptions of the loss distribution and therefore relying too heavily on the underlying data (Nilsson, 2013). 2.3 Parametric approach The parametric approach to estimate VaR is done by using probability curves and fitting them to the data, consequently deriving the VaR estimate from the probability curve. The main assumption behind this approach is that market volatility, or in other words price changes, follows a probability distribution 11

12 curve such as the bell- shaped normal distribution or the student s t- distribution curve. The main drawbacks of using this approach are related to the assumption that the market returns are normally distributed; while empirical evidence has shown that this is not the case. Mandelbrot (1963) also stresses the fact that the parametric models disregard the fact that financial returns are not identically and independently distributed (IID). In other words, high returns are usually followed by high returns, and low returns are usually followed by low returns, a phenomenon called volatility clustering Normal Distribution There has been criticism of VaR since the financial crisis of 2008 where the U.S. housing market and global financial markets collapsed. The criticism is related to the presence of fat tails in financial returns distribution (Olson & Desheng, 2013). As discussed earlier, financial returns are rarely normally distributed leading to VaR estimated under normal distribution could be either under- or overstated (Zhang & Cheng, 2005). This is due to the fact that, under normality, it allows for financial losses larger than the initial capital investment, wiping out more than the initial investment (Dowd, 2002). Although, the main attraction of using the normal distribution is because there are only two independent factors to consider, the mean,!, and the standard deviation,! (Dowd, 2002).!"#! =! +!!! 2.1 where!! is the critical value corresponding to the confidence level!. N(0,1) 95% VaR Figure 2.1 VaR under the normal distribution 12

13 Under a zero- mean, and standard deviation equal to 1 normal distribution curve VaR is - 1,64. In this sample it is the 95 th percentile largest loss shown above where the grey area starts. However, it is important to stress that VaR does not estimate how big the loss can be, but rather the smallest loss given the confidence level Student s t- distribution The student s t- distribution is rather simple and is, in contrast to the normal distribution curve, better suited for explaining financial returns because of it s fat tails and positive excess kurtosis. Because of the shape of the student s t- distribution it has been criticised since it does not capture the asymmetrical distribution of financial returns. Although, empirical results have shown that when using higher confidence levels, the student s t- distribution gives better VaR estimates than when derived from the normal distribution (Lin & Shen, 2006). Figure 2.2 Student s t- distribution under different degrees of freedom (Sage, 2012) 13

14 In order to estimate VaR using the t- distribution we need to consider the following equation:! =!(!!!)! =!!!!!!!!!! 2.2!"#!! =! +!!!!!!!,! 2.3 where,! is the degrees of freedom derived from using a maximum likelihood (ML) estimation based on the probability density function.!!,! is the!- quantile for the distribution, while the factor (! 2)! is interpreted as a scaling of! due to excess kurtosis (Dowd, 2002). The degrees of freedom can also be derived by using the Excel function KURT()+3 to get the kurtosis on the sample data. There has been extensive research on the topic on how many degrees of freedom should be used in order to get the most accurate fit of the student s t- distribution to the financial returns. Platen and Sidorowicz (2007) mentions some of the conclusions made by Markowitz and Usmen (1996a, 1996b), Hurst and Platen (1997), and Fergusson and Platen s (2006) ML estimation which showed that the degrees of freedom that gave the best fit was 4,5, 3,0-4,5, and 4,0 respectively. The underlying data in all these research papers were S&P 500 or a world stock index, whose constituent weights were determined by market capitalization, and considered different currency denominations of such an index (Platen & Sidorowicz, 2007). 2.4 Non- Parametric Approach The non- parametric approach, in contrast to the parametric approach, does not rely on strong assumptions of the return distribution. The main goal of these approaches is to let the data speak for itself as much as possible, under the assumption that the future will be similar to the past. In other words, the non- parametric distribution relies on the past empirical distribution of returns, rather than using a theoretical distribution curve (Abad et al., 2013) Historical Simulation According to Pérignong and Smith s (2010) international survey, Historical Simulation (HS) along with the similar method, Filtered Historical Simulation, are by far the most established methods for estimating VaR at commercial banks. 14

15 To forecast the conditional quantiles of financial returns, the HS approach uses unconditional quantiles of financial data (Escanciano & Pei, 2012). In its simplest form, the HS approach takes the largest loss from the sample window that intersects with the lower 5% of returns and estimates this as VaR on a 95% confidence level. Given a sample of 1000 observations, the VaR will be the 51 st largest loss. The reason for taking the 51 st and not the 50 th largest loss lies in how VaR is defined.!"#!! =!"!!: Pr! >! 1! 2.4 Pr (! >!"#!! = 1! 2.5 In essence, the probability of a loss larger than the VaR is equal to 1!, in this case 95%, therefore, the 51 st largest loss is estimated as VaR (Dowd, 2002) Age Weighted Historical Simulation A more sophisticated relative to the HS approach is the Age Weighted Historical Simulation (AWHS). One drawback of the HS, which AWHS is trying to overcome, is the fact that all observations in the sample are given the same weight. In other words, an observation will affect the VaR estimate in the same way, no matter if it s the first day or the last day in the sample window. Depending on the sample window, market characteristics may have changed drastically, e.g. during volatility clustering, where an old observation will contribute in a negative way to the VaR estimate. The HS approach depends on the assumption that each observation in a sample period has the same probability to happen again and is independent from other observations over time (IID), which in the end can create ghost effects (Dowd, 2002). However, the AWHS approach adopted by Boudoukh et al. (1998) overcomes this issue. They took a hybrid approach between the HS and exponential smoothing (EXP) approach, but since the EXP approach is parametrically driven, it assumes normality. As discussed earlier, financial returns are rarely normally distributed, but rather display fat tails, excess kurtosis and unstable correlations, hence the rise of the AWHS approach. 15

16 The idea is simple, ranking the largest losses first and assign weights to each of the financial returns, based on their age in the sample window. The next step is to add up the cumulative weights until it breaches the confidence level of choice. The weights are mathematically explained below (Boudoukh et al., 1998):!! = (1!) (1!! )!! =!!!!! =!!! =!!!!!! =!!!!!! 2.6 where!! is the probability of the latest observation and!! is the probability of the oldest observation in the sample window.! is the irrelevance factor, i.e. how fast the observations will decrease in probability. A! close to 0 will make older observations irrelevant quicker than a! closer to 1 (Boudoukh et al., 1998). In order to adapt the model for volatility clustering, a reasonable! will be used when conducting the analysis. 2.5 Semi- Parametric Approach As one could imagine, the semi- parametric approaches combines the parametric approach with the non- parametric approach. The most established semi- parametric methods are volatility weighted historical simulation, CaViaR and the extreme value theory method (Abad et al., 2013), while this thesis will focus on volatility weighted historical simulation Volatility Weighted Historical Simulation In addition to the AWHS approach, there are also other ways of weighting sample data. One such approach is the Volatility Weighted Historical Simulation (VWHS) originally developed by Hull and White (1998). The basic concept of this approach is to weight the observations depending on current market volatility. As an example, if the current market volatility is 2%, while 20 daily observations earlier (i.e. one month ago), the market volatility was 3%. The one month old data overstates the price changes in the market compared to what would be 16

17 expected in the current state of the market. The same applies to when the volatility was lower but the current state of the market implies a higher volatility, i.e. the older data understates the expected volatility (Hull & White, 1998). The main motivation of using this approach is due to the volatility clustering phenomena mentioned earlier. The main advantage of using the VWHS instead of the non- parametric approaches HS and AWHS is that the VWHS takes account of volatility changes in a natural and direct way. In contrast, the HS approach completely ignores changes in volatility, while it is not fully incorporated in the AWHS approach. The VWHS approach also enables the VaR estimates to exceed the largest loss in the sample period. This event happens when there are moments of high volatility since the observations in the sample period are scaled upwards (Dowd, 2002). Empirical evidence has also shown that the VWHS approach is producing better VaR estimates than the HS and AWHS approaches (Hull & White, 1998). The scaling of losses are explained mathematically below: l! =!!!!!! l! l! =!!!!!! l! l! =!!!!!! l! where!!,!!,,!! are the volatilities of each and every observation in the sample period and!!!! is the forecasted volatility of the next observation, consequently estimated using an exponentially weighted moving average (EWMA) model.!!!!! = 1! 1!!!!!!! is a fixed constant equal to 0,94 used by RiskMetrics and the error term,! is initially set to 0.!!!!!!!

18 2.6 Backtesting VaR The validity of VaR models is usually measured on their ability to forecast reliable VaR estimates. Since VaR was introduced, several statistical tests have been developed in order to estimate the validity of the VaR estimates, where the two most established ones being the Kupiec s (1995) likelihood ratio test and the independence test of Chrisoffersen (1998) Kupiec Test In short, the Kupiec s test is a way of measuring the number of allowed exceptions. The backbone of the test are formal statistical models in order investigate the accuracy of VaR models. Even though Kupiec (1995) found that the main drawback of the test is that it requires large samples to function properly, it is still the most established test and is widely used. The test is formed under Kupiec s null hypothesis where the expected number of violations of the VaR estimate should follow a binominal distribution.!! :! =! =!! 2.11 Where T is the number of observations in the sample period and x is the number of violations under a certain confidence level. The test s intention is to test if the observed number of violations,!, is significantly different from the expected number of violations,!. In other words, a VaR break (VB) occurs when a larger loss than the VaR estimate is observed. It has been shown that the hypothesis is best tested under a likelihood ratio (LR) test (Kupiec, 1995).!" = 2 ln 1!!!!!! 1!!!!!!!! ~!! (1) 2.12 LR is!! (chi- squared) distributed with one degree of freedom (d.o.f.). In the case where LR is exceeding the critical value of the!! - distribution, the null hypothesis will be rejected. 18

19 2.6.2 Christoffersen Test The Kupiec test does not take in to account conditional coverage in order to detect violations of the independence property of an appropriate VaR estimate. Due to this fact a large number of tests have risen, of which the Christoffersen test is one of those. The Christoffersen test is testing whether or not a violation at t is dependent on if there was a violation at t- 1 (Campbell, 2005). A conditional coverage test is used to investigate whether or not the underlying model is estimating correct VB frequency and if these VB s are independent from each other. The Christoffersen test also applies a!! - distribution using a LR test.!"!"# = 2 ln! 1!!!!!!"!!!!"!!!!! 1!!!!!!!"!"!" 1!!!!"!!!!! ~!! 1!! where!!,! is the number of days where state! occurred the day before state!;!!" =!!"!!!!!!"!!! =!!!!!"!!!!!! =!!"!!!!!!!!!!"!!!"!!!! 2.13 However, there are a few shortcomings of this method. The existence of numerous ways where the independence property could be violated is the main drawback. This is due to the shortcoming in the alternative hypothesis where it can t be distinguished in which way the hypothesis is being violated. As for the Kupiec test, the Christoffersen test is being validated using a p- value for the chosen confidence level if the hypothesis can be rejected or not (Campbell, 2005). 19

20 3. Methodology In this section of the thesis the different methodical approaches used will be discussed. The nature of the thesis will be presented firstly, followed by a presentation of each and every index considered in the thesis. After this, a short summary of key statistics will be discussed. Lastly, there will be a presentation of what steps have been taken in order estimate VaR using the different approaches. 3.1 Quantitative Approach The nature of VaR almost forces the study to take a quantitative approach since the data set used are based on almost 20 years of historical data from five different stock indices. A quantitative approach is a research strategy relying on the quantification in collection and analysis of data. Quantitative researchers measure their result in contrast to qualitative researchers (Bryman & Bell, 2011). These measurements can often be viewed as the truth, but in the case of VaR estimation they should not. It is important to stress that the sample data and sample window may produce biased results, and even if the study is applicable, it may not give true results for other sample data. In order to get to a conclusion, a large amount of calculations have been made on the historical data to estimate VaR. Therefore, for this thesis, a quantitative approach is the best way of conducting a reliable analysis in order to draw relevant and reliable conclusions. 3.2 Deductive Approach Firstly, the main purpose of this thesis is not to invent a new theory, but solely to test and analyse different established VaR models, therefore the thesis will take a deductive approach. As Bryman and Bell (2011) points out, the process of deductive research starts with the theory, which is then applied to the observations/findings. However, there are some drawbacks when a deductive approach is taken. (1) There could be some new findings or research published by others, before the researcher s findings (thesis) has been published. (2) The data set may become irrelevant for the theory, which is only apparent when the theory is eventually applied. (3) There are no guarantees that the data is suitable for the purpose of the research (Bryman & Bell, 2011). The idea is to apply the 20

21 theoretical approaches on the empirical evidence and draw conclusions that will either strengthen or weaken the theory. To justify the thesis taking a deductive approach, even though the approach has its drawbacks, the following comments may be worth taken in to consideration. (1) Due to the time period of which this thesis is conducted, and the amount of research already made on similar topics, there is a limited risk that new findings will make the thesis irrelevant. (2) In order to conduct VaR estimates there is a need of financial data, and this is what the different VaR approaches uses when estimating VaR figures. (3) This is in fact the purpose of the thesis, the data sample contains moments of high volatility, which may be unsuitable for estimating VaR, but in order to make sure which VaR approach gives the best estimates during volatility clustering, this is a necessity. However, this could also lead to inconclusive results if none of the approaches are suitable. 3.3 Reliability and Validity To ensure that the results and conclusions drawn in this thesis are valid, the historical data that have been used are based on public information and enables reproduction of the conducted tests. There is also extensive research made on the topic of VaR, which ensures the reliability, and validity of the thesis. It is however important to stress that VaR is an estimation of potential future loss and not the absolute truth. The fact that historical data of five stock indices does not tell the entire truth of what has happened in the global financial markets the last decade is also worth considering. The correlation between the stock index and a specific stock is not a perfect match, there is a possibility that a stock price can move in the opposite direction of the market or not even move at all. There are also other asset classes like FX, rates and fixed income etc. that is not considered in this thesis. An investor, or a financial institution that holds large portfolios of different assets may experience different VaR estimates and volatility than for the five chosen indices. 21

22 3.4 The Indices In order to investigate how appropriate the different VaR approaches are, 5 different indices has been chosen, all of which are highly diversified and geographically different. The selection has been made in order to create a robust and reliable analysis of the VaR models. Below there will be a quick introduction to each index and a graph that explains the market conditions from to For each of the indices, there is significant volatility clustering throughout the observation period. The largest and highest of which happened during the financial crisis of , which is also coherent in the logarithmic return distribution showing large positive and negative returns OMX Stockholm 30 The OMXS 30 index is a list of the 30 most traded stocks on the Stockholm Stock Exchange and is an index that is priced based on market weights. It was established on the 30 th of September 1986 with a base level of 125 and is denoted in Swedish Krona (SEK). The list of stocks on the OMXS 30 is revised twice a year (Bloomberg, 2014). 40,00% 30,00% 20,00% 10,00% 0,00% - 10,00% - 20,00% Log- return MVol QVol Figure 3.1 The graph above show the Logarithmic return, Monthly volatility and Quarterly volatility (in %) on the OMXS 30 index. 3 There are many perceptions on when the recent financial crisis actually started. The start date of the recent financial crisis is set to the 15th of September 2008, the day when Lehmann Brothers was filed under bankruptcy. 22

23 3.4.2 Standard & Poor s 500 The S&P 500 is one of the most famous stock indices in the world and is a market capitalization weighted index with 500 different stocks. It was established to measure the performance of U.S. economy by looking in to the market value of companies in all the major industries. It was originally established with a base level of 10 for a period and is denoted in U.S. Dollar (USD) (Bloomberg, 2014). 40,00% 30,00% 20,00% 10,00% 0,00% - 10,00% - 20,00% Log- return MVol QVol Figure 3.2 The graph above show the Logarithmic return, Monthly volatility and Quarterly volatility (in %) on the S&P 500 index NIKKEI of the highest rated Japanese companies constitute the NIKKEI 225 index. It is a price- weighted index with companies listed on the First Section of the Tokyo Stock Exchange. NIKKEI 225 was first established on the 16 th of May 1949 with a base level of 176,21 Japanese Yen (JPY) (Bloomberg, 2014). 40,00% 30,00% 20,00% 10,00% 0,00% - 10,00% - 20,00% Log- return MVol QVol Figure 3.3 The graph above show the Logarithmic return, Monthly volatility and Quarterly volatility (in %) on the NIKKEI 225 index. 23

24 3.4.4 Deutscher Aktie IndeX 30 DAX 30 is a total return index consisting of 30 blue chip stocks traded on the Frankfurt Stock Exchange. In order to do the index calculation, DAX 30 is using free float shares. It was established on the 31 st of December 1987 with a base value of 1,000 (Bloomberg, 2014). 40,00% 30,00% 20,00% 10,00% 0,00% - 10,00% - 20,00% Figure 3.4 The graph above show the Logarithmic return, Monthly volatility and Quarterly volatility (in %) on the DAX 30 index FTSE 100 Similar to the S&P 500, the FTSE 100 is also a capitalization weighted index, though, only consisting of 100 of the highest capitalized stocks on the London Stock Exchange. FTSE 100 uses investibility weighting in order to calculate the index level. It was originally established on the 30 th of December 1983 with a base value of 1,000 and is denoted in British Pound (GBP) (Bloomberg, 2014). 40,00% 30,00% 20,00% 10,00% 0,00% - 10,00% Log- return MVol QVol - 20,00% Log- return MVol QVol Figure 3.5 The graph above show the Logarithmic return, Monthly volatility and Quarterly volatility (in %) on the FTSE 100 index. 24

25 3.5 Data The data of the five indices described above will be used for the empirical evaluation and has been sourced from Thomson Reuters DataStream. The data contains daily closing prices from the 1 st of October 1993 to the 1 st of October It should however be stressed that the time period effectively used will stretch from 1 st of October 1994 to 1 st of October 2013 due to the computational requirements of the VaR approaches. Effectively, 4,957 VaR observations have been taken in to account when making the empirical evaluation, but due to the nature of this thesis, the focus will lye on the period from 2007 to 2012, during the recent financial crisis. In the event where one or more indices have been closed due to e.g. holidays, the same closing price as the day before has been assigned in order to have consistency between the indices. The logarithmic returns have been calculated by using the formula ln(!!!!!! ) where! is the closing price at time!. It should also be stated that the data does not capture intra- day volatility and this has been disregarded due to the already large magnitude of observations. 3.6 Descriptive Statistics In order to give a view of the data, some preliminary statistics, such as mean, standard deviation, kurtosis, skewness and Jarque Bera 4 (JB) test, are presented in table 3.1. As discussed earlier in the literature review, empirical evidence has shown that financial data are seldom normally distributed. Bearing this in mind, we expect that none of the return distributions for the indices to be normally distributed. In table 3.1 we see that normality is strongly rejected under the JB- test. As well as in the case of non- normality, the data shows significant excess kurtosis, or in other words, the data exhibits leptokurtic features with fat tails. The most different, and also the most interesting, statistic between the five indices is the skewness. OMXS 30 shows a positive skewness meaning that the return distribution is slightly skewed to the right, while the rest of the indices show smaller negative skewness i.e. the return distributions are skewed to the left. 4!" =!!!! +!!! 3! ~!! (2), where skewness, S = 0 and kurtosis, K = 3 under normality. 25

26 OMXS30 S&P500 NIKKEI225 DAX30 FTSE100 Mean 0,0003 0,0003 0,0000 0,0003 0,0001 Std dev 0,0150 0,0120 0,0149 0,0149 0,0117 max 11,02% 10,96% 13,23% 10,80% 9,38% min - 8,53% - 9,47% - 12,11% - 8,87% - 9,27% Kurtosis 6,79 11,54 9,12 7,53 9,12 Skewness 0,08-0,24-0,29-0,12-0,16 Jarque Bera 3 081, , , , ,01 Number of observations Table 3.1 The table shows descriptive statistics for the 5 indices over the time period to VaR Estimates The following section will give a detailed description on how the VaR estimates were produced for each VaR approach. This will enable the reader to reproduce and recreate the same approach taken to conduct a similar study. Three different rolling sample windows has been used, 63 days (one quarter), 252 days (one year) and 1000 days (approx. four years). The data, i.e. daily closing prices, has been converted in to daily logarithmic returns, which has been used in order to produce the VaR estimates Normal Distribution In order to produce the VaR estimates under the normal distribution approach the first step is to assign a volatility,!, to each and every observation. This was made by using Excel s function STDEV.S(). Depending on the sample window, the previous X 5 observations were included in the formula. The same procedure was made in order to calculate the mean,!, using AVERAGE() in Excel. Since the thesis only considers a 95% confidence level, the z- value,!!, under normal distribution is equal to - 1,64. The calculations were then repeated throughout the period of 2007 to Student s t- Distribution Under the student s t- distribution the first step was to calculate the degrees of freedom. The degrees of freedom were estimated by using the KURT() formula in Excel on the period from to Under this presumption, the 5 Sample window - 63, 252 or 1000 observations 26

27 degrees of freedom ranged from 4,7 to 5,6. This means that they are significantly different from what previous researchers have found the most appropriate match of around 4,0 degrees of freedom described in the literature review. The student s t- distribution was also tested under degrees of freedom equal to 4,0 (A short analysis can be found in section 5). The critical t- value,!!,!, under 5,0 degrees of freedom is equal to - 2,015. The mean and volatility was calculated by using the same method as in the normal distribution approach Historical Simulation The historical simulation approach is a relatively simple approach and is fairly straightforward. Depending on the sample window, the X th largest loss in the sample window is the VaR estimate. X was found by taking the 5 th percentile on the number of observations in the sample window, e.g. if the sample window contained 252 observations, the 13 th largest loss (252 5% = 12,6) was estimated as VaR Age Weighted Historical Simulation In order to estimate VaR in an efficient matter, and including a rolling sample window, a VBA macro was written. The VBA transcript can be viewed in the Appendix. However, the steps the VBA macro takes when estimating VaR using the AWHS approach on a rolling sample window are the following. (1) Copy the X 6 previous observations and assign the weights depending on! which in this case was set to 0,99 (Formula 2.6). (2) Sort the losses in an ascending order, i.e. from smallest to largest. (3) The VaR estimate is the first loss where the sum of the sum of the weights is larger than 5%. (4) The VBA code then repeated this by taking the next day s X previous observations until reaching the end of the sample period Volatility Weighted Historical Simulation Similarly as for the AWHS VaR estimates, a VBA macro was written and the transcript can be found in the Appendix. The first step was to forecast the! volatility,!!!!, using an EWMA model for each day in the observation period! (Formula 2.8). When!!!! is estimated for each and every observation in the 6 Sample window 63, 252 or 1000 observations 27

28 sample period the next step is to scale the losses according to formula 2.7. The scaled losses are then sorted in an ascending order similarly as in the case of AWHS. The VaR estimate is the X th largest loss in the sample window e.g. the 13 th largest loss in a sample window of 252 observations. 28

29 4. Empirical Evidence In this chapter the empirical findings will be presented. In order to avoid confusion, each VaR approach and backtest will be presented separately in order to get a better understanding of the results. When presenting the backtesting results, a green number means that the number of VB s is accepted under the Kupiec test, whereas a red underlined number means that it is rejected. In section 4.6, results from the Christoffersen test of independence will be presented, using the same formatting rule as for the Kupiec test. 4.1 VaR under Normal Distribution Table 4.1 presents the number of VB s under normal distribution with a rolling sample window of 63 observations. Normal distribution, under this approach, constantly underestimates VaR, with an exception in 2009, which shows satisfactory VaR ratios (VR s 7 ). Recalling figures 3.1 to 3.5, volatility was constantly at a low decreasing level in 2009 relative to the sample period of , consequently causing fewer VB s. Index OMXS 30 S&P 500 NIKKEI 225 DAX 30 FTSE 100 Year VB VR VB VR VB VR VB VR VB VR ,35% 25 9,92% 19 7,54% 20 7,94% 16 6,35% ,75% 22 8,73% 17 6,75% 21 8,33% 20 7,94% ,78% 12 4,76% 13 5,16% 12 4,76% 13 5,16% ,16% 18 7,14% 14 5,56% 14 5,56% 14 5,56% ,94% 17 6,75% 15 5,95% 22 8,73% 19 7,54% ,95% 15 5,95% 13 5,16% 11 4,37% 16 6,35% ,82% 109 7,21% 91 6,02% 100 6,61% 98 6,48% Table 4.1 VaR breaks and VaR ratio under normal distribution with a sample window of 63 observations. In contrast to the sample window using 63 observations, the 252 observations sample window clearly overestimates VaR in 2009 and 2012 where volatility was lower than previous periods. This clearly displays VaR s attribute of slow adaptability to changes in market conditions. However, in 2010 the VR show satisfactory figures throughout all indices, possibly due to calm volatility changes from 2009 to Out of all indices, the one index showing satisfactory VR s throughout the whole sample period, with an exception in 2009, is the NIKKEI 225. In figure 3.3, NIKKEI 225 show relatively high and stable volatility with few 7 VR = 1 (252 n) / 252, where n = no. of VB s in a year 29

30 exceptions during the sample period, most probably causing an overall satisfactory VR for the sample period. Index OMXS 30 S&P 500 NIKKEI 225 DAX 30 FTSE 100 Year VB VR VB VR VB VR VB VR VB VR ,73% 29 11,51% 19 7,54% 22 8,73% 22 8,73% ,71% 33 13,10% 29 11,51% 27 10,71% 26 10,32% ,59% 4 1,59% 3 1,19% 5 1,98% 3 1,19% ,16% 11 4,37% 12 4,76% 10 3,97% 13 5,16% ,92% 23 9,13% 6 2,38% 30 11,90% 23 9,13% ,59% 4 1,59% 7 2,78% 4 1,59% 4 1,59% ,28% 104 6,88% 76 5,03% 98 6,48% 91 6,02% Table 4.2 VaR breaks and VaR ratio under normal distribution with a sample window of 252 observations. When applying a rolling sample window of 1000 observations the results become somewhat worrying. In table 4.3 there is evidence that slow adaptability can both over- and underestimate VaR. VR s show large spreads in VB s from year to year, with a clear overestimation in 2010 and 2012 and underestimation in 2007 and Another worrying fact is the lack of VB s in 2012 for NIKKEI 225 and FTSE 100, and only one VB for S&P 500. The VR, ranging from 5.69% to 7.94%, for the overall sample period show that VaR is overestimated through all indices and not displaying a satisfactory number of VB s. Index OMXS 30 S&P 500 NIKKEI 225 DAX 30 FTSE 100 Year VB VR VB VR VB VR VB VR VB VR ,73% 34 13,49% 18 7,14% 19 7,54% 28 11,11% ,86% 56 22,22% 49 19,44% 46 18,25% 55 21,83% ,56% 15 5,95% 11 4,37% 17 6,75% 14 5,56% ,59% 7 2,78% 5 1,98% 5 1,98% 6 2,38% ,16% 7 2,78% 3 1,19% 16 6,35% 8 3,17% ,59% 1 0,40% 0 0,00% 4 1,59% 0 0,00% ,75% 120 7,94% 86 5,69% 107 7,08% 111 7,34% Table 4.3 VaR breaks and VaR ratio under normal distribution with a sample window of 1000 observations. The normal distribution approach only gives satisfactory VaR estimates during moments of low or constant volatility, expectedly, due to the normal distribution s lack of fat tails and leptokurtic features. The results show that the normal distribution approach under a sample window of 252 observations is superior to the other sample windows when estimating VaR over the whole sample period. However, when inspecting the fact that the sample window of 252 observations has a wide spread of VB s from year to year, the sample 30

31 window of 63 observations clearly gives the most satisfactory VaR estimates during volatility clustering Backtest of VaR under Normal Distribution In table 4.4 the results from the Kupiec s test of unconditional coverage are presented. The worst fit seems to be the sample window of 1000 observations. It has far more VB s than what is acceptable and the difference in VB s from year to year is widely spread. Consequently, rejecting a normal distribution VaR model using a sample window of 1000 observations, where only the NIKKEI 225 index is showing an acceptable number of VB s over the whole sample period. Even though the total VB s for the 225 observations sample window (464 VB s) are less than for the 63 observations sample window (486 VB s), it seems to be a worse model. The only year where the sample window of 252 observations are showing satisfactory results are 2010, while when using a sample window of 63 observations gives satisfactory results from year to year with minor exceptions. Index OMXS 30 S&P 500 NIKKEI 225 DAX 30 FTSE 100 Year MIN MAX Target 63D 63D 63D 63D 63D Year MIN MAX Target 252D 252D 252D 252D 252D Year MIN MAX Target 1000D 1000D 1000D 1000D 1000D Table 4.4 Kupiec s Test for 95% VaR under normal distribution. 31